A depth-integrated non-hydrostatic model for nearshore wave modelling based on the discontinuous Galerkin method. (15th July 2021) - Ran, Guoquan; Zhang, Qinghe; Li, Longxiang
Abstract: In this study, a convergent two-dimensional depth-integrated non-hydrostatic shallow water model is discretized using the discontinuous Galerkin (DG) method. This model can account for the dispersive effects by including a non-hydrostatic pressure component and can be used for the simulation of weakly dispersive water waves. Additionally, this model is based on the fractional step method. In the first step, the traditional nonlinear shallow water model plus a transport equation for vertical momentum is discretized by the DG method, and the resulting semi-discrete system is evolved in time by the explicit third-order strong-stability-preserving Runge-Kutta (SSPRK3) method to obtain the provisional solution. In the second step, the provisional solution is corrected by satisfying a divergence constraint for the velocity. The latter step is conducted after application of the DG discretization to an elliptic equation regarding the non-hydrostatic pressure. Both theoretical and experimental tests are carried out to validate the proposed model. The results indicate that for smooth problems, our model can arrive at a given error tolerance by using a mesh with fewer degrees of freedom (DOFs) at the cost of a smaller CPU time when a higher approximation order is adopted and can properly simulate wave run-up, diffraction, refraction, and focusing. Highlights: A convergent depth-integrated non-hydrostatic shallow water model based on the discontinuous Galerkin method is presented. For smooth problems, with a higher approximation order, this model can arrive at an error tolerance at the cost of a smaller CPU time. With the convective acceleration terms included, this model can simulate wave motion better than model without those terms.